A SERIES FOR THE COLLISION PROBABILITY IN THE SHORT-ENCOUNTER MODEL
R. Garcı́a-Pelayo ∗
J. Hernando-Ayuso†
Departmento de Fı́sica Aplicada
Escuela T. S. de Ingenierı́a Aeroespacial
Plaza del Cardenal Cisneros 3,
Universidad Politécnica de Madrid
Madrid 28040, SPAIN
Graduate School of Engineering
Department of Aeronautics and Astronautics
The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, JAPAN
This paper is a version without tables of an article which will appear
in the Journal of Guidance, Dynamics and Control [1]
ABSTRACT
A series to compute the collision probability of two spheres under the assumptions of short encounter has been derived. It is valid for both Gaussian
and non-Gaussian distributions of the position. In the particular case of a
Gaussian distribution the use of Hermite polynomials yields a simple form
for the series. A region of practical interest has been carefully defined, and
a sampling set of 244 cases was chosen. On this sampling set a comparison
between the new series and previous algorithms has been performed for the
Gaussian case. The presented series is faster than any other algorithm in every case. Numerical evidence suggests that if the series for the Gaussian case
is truncated when the last term is smaller than the computed probability times
a tolerance of 0.1, then the last term is an upper bound for the error. This article also presents very strong evidence for the case that the first two terms
of the series are sufficient for the computation of the probability of collision
and the absolute value of its second term is an upper bound for the error made
when using it.
Index Terms— Space debris, collision probability, evasive maneuver,
non Gaussian, fast computation
1. INTRODUCTION
The solution to the problem of space debris requires cleaning up the space
around the Earth [2], especially the low Earth orbits, and enforcing laws that
require space vehicles to take care of their own removal. The post-mission
disposal is performed either by transfering to a graveyard orbit, or by descending to a low altitude orbit. This can be achieved either by saving enough
fuel to go down to a low altitude or by passive means, such as electrodynamic
tethers [3].
An active satellite or space station may be impacted by another body
and suffer total or partial mission failure, or even fragment into thousands
of pieces as it has happened in the past (see for instance the 2009 IridiumCosmos collision [4]). Derelict satellites and rocket bodies, and other debris
cannot be removed from crowded orbits until active debris-removal technology is mature. In the mean time in order to avoid collisions, the best that one
can do is a collision avoidance maneuver when another body threatens to hit
the satellite. Performing a collision avoidance maneuver costs fuel, and one
has to compute the collision probability first and then do the evasive maneuver if it surpasses certain threshold (i. e. between 10−5 and 10−3 ) [5, 6, 7].
∗ Supported by the research project entitled “Dynamical Analysis, Advanced Orbit Propagation and Simulation of Complex Space Systems”
(ESP2013-41634-P) of the Spanish Ministry of Economy and Competitiveness
† Supported by a scholarship for graduate school students of the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of the
Japanese Government
These kinds of maneuvers are performed several times during the lifetime of
a satellite: as an example Envisat executed 4 maneuvers in 2011 [8].
The computation of the collision probability between two encountering
bodies has been the subject of research since at least 1992 [9]. The usual
assumptions in this computation are that the relative motion is a straight line
within the region where the collision could take place, that their positions are
Gaussianly distributed and that the objects are spheres. Akella and Alfriend
[10] described a solution of this problem using an integral over space and
time to take into account the relative motion of the bodies. Another similar,
but less known method is the one of Khutorovsky et alii [11], which is the
only precedent that we know of the approach presented in this article.
However, under the assumptions stated above the problem can be reduced to a time-independent, two-dimensional spatial integral. This is
achived by essentially projecting the problem on a plane. In [9] Foster and
Estes presented a numerical method to compute the collision probability
p with a fixed spatial discretization. A finer discretization increases the
precision of the calculation, but at a high computational cost. Patera [12, 13]
solved the problem in polar coordinates doing a first analytic integration
along the radial direction, followed by a numerical integration over the angle. The accuracy of the result can be improved using a tighter tolerance in
the numerical integration. Chan [14, pp. 63 - 97] proposed an analytic power
series that solves the isotropic problem (this is, the relative position uncertainty is the same in every direction). The anisotropic case is approximated
using the previous result. While arbitrary accuracy can be achieved in the
isotropic case by taking additional terms, the error made by approximating
the general case by the isotropic one will not vanish. Another possibility is
to discretize the integral and write it as a power series, which is the approach
used by Alfano [15]. Note that Alfano had to adjust heuristically one of
the terms, so this is not a purely analytical method. There is a very recent
work of Serra et alii [16] which is fully analytic. Some of its features will be
mentioned in sections 6 and 7.
In section 2 of this article we state the problem. In section 3 we find
an expansion valid for a general probability density function (pdf) ρ of the
relative position. When ρ is a Gaussian we show in section 4 that the expansion takes a simple form which involves Hermite polynomials. In sections
5 through 7, for the Gaussian case, we compare the performances of our algorithm and those of other authors and find that our algorithm is the fastest.
In section 6 we show that, for all practical matters, our expansion can be reduced to only two terms, the second of which is also a bound for the error.
Conclusions are stated in section 8.
We note that the probability of collision computed here or by other
authors can be applied to any two objects that may collide whenever the
three above mentioned assumptions hold, for example in artillery or other
instances.
2. STATEMENT OF THE PROBLEM
Let there be two spheres of radii R1 and R2 whose centers are independently
distributed according to the pdf’s f1 (r, t) and f2 (r, t), respectively. The as-
sumption that the objects are spheres is a conservative approximation if these
spheres are taken as the spherical envelopes of the bodies that might collide.
In a more exact approximation the objects could be substituted by their convex envelope and then consider Minkowski sums of their convex envelopes,
which are easier to compute than the Minkowski sum of non convex bodies
[17, Chapter 3: Minkowski addition].
There is an encounter plane (also called “b-plane”) that contains the expected position of the two spheres at the expected time of closest approach,
tc . This plane is perpendicular to the expected relative velocity at that time.
In order to define this formally, we denote the expected value by h i, that
is
Z
hri1,2 (t) ≡
d3 r f1,2 (r, t) r.
(1)
Then tc is the root of the equation
d
(hri2 (tc ) − hri1 (tc ))2 =
dt
(2)
d
(hri2 (tc ) − hri1 (tc ))
(hri2 (tc ) − hri1 (tc )) = 0.
dt
The encounter plane is perpendicular to the expected value of the relative
d
velocity dt
(hri2 (t) − hri1 (t)) (tc ) and contains the centers of the two
spheres at the time tc , hri1 (tc ) and hri2 (tc ).
We denote by ρ1 and ρ2 the projections of f1 (r, t) and f2 (r, t) on
the encounter plane, that is, the marginal pdf’s resulting from integrating the
original pdf’s along the directions parallel to the expected value of the relative
velocity. This reduces the three-dimensional collision/non-collision problem
of two spheres to the two-dimensional overlap/non-overlap problem of two
circles, which are the projection of the spheres. This simplification holds if
the relative motion does not deviate significantly from a straight line for the
duration of the encounter, and if the uncertainty of the relative velocity is
negligible. This is called the short-encounter model [9, 10, 12, 14, 15, 16,
18, 19].
There is an approximation for the collision probability of these two
spheres, which is [11]
Z
p ≈ π(R1 + R2 )2 d2 r ρ1 (r)ρ2 (r).
(3)
Uniform convergence and power series]. Then,
Z
∞
X
X
1
∂ 2i ρ2 (r1 )
p=
dr1 ρ1 (r1 )
(2i)!
∂(r1 )j1 · · · ∂(r1 )j2i
i=0
j1 ,...,j2i =x,y
Z
dr2 (r2 − r1 )j1 · · · (r2 − r1 )j2i .
|r2 −r1 |<R1 +R2
R
The integrals |r −r |<R +R dr2 (r2 − r1 )j1 · · · (r2 − r1 )j2i are of
2
1
1
2
R
the form |r −r |<R +R dr2 (x2 − x1 )2a (y2 − y1 )2b , where a and b
2
1
1
2
are integers and the exponents 2a and 2b are even, otherwise the integral
vanishes. They can be represented using [21, integral 3.621.5])
Z 2π
(2a − 1)!!(2b − 1)!!
π.
(7)
dϕ cos2a ϕ sin2b ϕ =
2a+b−1 (a + b)!
0
Indeed,
Z
dr2 (x2 − x1 )2a (y2 − y1 )2b =
|r2 −r1 |<R1 +R2
Z
R1 +R2
d` `2i+1
Z
2π
3. THE EXPANSION
The probability that two circles of radii R1 and R2 whose centers are independently distributed according to ρ1 and ρ2 overlap is
Z
p=
(R1 + R2
− 1)!!(2b − 1)!!
π=
2i + 2
2i−1 i!
(R1 + R2 )2i+2
(2a − 1)!!(2b − 1)!! π.
2i (i + 1)!
When j1 , . . . , j2i take the values x and y, the above integral appears
times. Therefore,
p=
∞
X
i=0
1
(2i)!
Z
dr1 ρ1 (r1 )
dr2 ρ2 (r2 ) =
|r2 −r1 |<R1 +R2
(9)
i X
i
∂ 2i ρ2 (r1 )
.
j ∂x2(i−j) ∂y 2j
j=0
When the two positions r1 and r2 are distributed according to the densities ρ1 and ρ2 , respectively, then the relative position ∆r ≡ r2 − r1 is
distributed according to the density
Z
Z
ρ(∆r) =
d2 r1 ρ1 (r1 )ρ2 (r1 +∆r) =
d2 r ρ1 (r−∆r)ρ2 (r). (10)
It is convenient to use the convolution product notation,
Z
f ⊗ g(r) ≡
dr00 f (r0 )g(r − r0 ),
(11)
for the above expression and write:
(4)
Z
dr2 ρ2 (r2 )
dr1 ρ1 (r1 ).
|r1 −r2 |<R1 +R2
If ρ2 is analytic, then it can be expanded in power series:
Z
Z
p=
dr1 ρ1 (r1 )
dr2
|r2 −r1 |<R1 +R2
∂ i ρ2 (r1 )
(r2 − r1 )j1 · · · (r2 − r1 )ji .
∂(r
)
1 j1 · · · ∂(r1 )ji
=x,y
X
j1 ,...,ji
2i 2b
i X
2i
∂ 2i ρ2 (r1 )
2j ∂x2(i−j) ∂y 2j
j=0
(R1 + R2 )2i+2
(2(i − j) − 1)!!(2j − 1)!! π =
2i (i + 1)!
∞ X
R1 + R2 2i+2
4π
2
(i
+
1)! i!
i=0
dr1 ρ1 (r1 )
(8)
Z
dr1 ρ1 (r1 )
Z
∞
X
1
i!
i=0
dϕ cos2a ϕ sin2b ϕ =
0
)2i+2 (2a
0
Z
In the above expression and henceforth, the domain of integration is always
R2 and the vectors are always in R2 , unless otherwise stated. The above approximation is good when ρ1 or ρ2 do not vary significantly over the distance
R1 + R2 .
(6)
(5)
R
When i is odd the integrals |r −r |<R +R dr2 (r2 −r1 )j1 · · · (r2 −
2
1
1
2
r1 )ji vanish due to the symmetry of the integration domain. We also suppose that the Taylor expansion of ρ2 about any point r1 converges uniformly,
so that the sum and integral signs may be commuted freely [20, Chapter 24:
ρ(∆r) = ((ρ1 ◦ (−1)) ⊗ ρ2 )(∆r),
(12)
where (ρ1 ◦ (−1))(r1 ) ≡ ρ1 (−r1 ). The function ρ(∆r) is further discussed in [22, 23].
Collision takes place when the distance between the centers of the
spheres is smaller than R1 + R2 , thus
Z
p=
d2 ∆r ((ρ1 ◦ (−1)) ⊗ ρ2 )(∆r).
(13)
|∆r|<R1 +R2
It follows that Eq. (9) will yield the same result for any other couple (ρ01 , ρ02 )
of pdf’s such that (ρ01 ◦ (−1)) ⊗ ρ02 = (ρ1 ◦ (−1)) ⊗ ρ2 . Furthermore,
the probability of collision depends only on the relative distance, so that Eq.
(9) will also yield the same result for any couple (ρ01 , ρ02 ) of pdf’s such that
(ρ01 (r1 ), ρ02 (r2 )) = (ρ1 (r1 + r0 ), ρ2 (r2 + r0 )), ∀r0 ∈ R2 . We are going
to use these two symmetries of ((ρ1 ◦ (−1)) ⊗ ρ2 )(∆r) to simplify Eq. (9).
R
A < a < b < C. The convolution of two 2-dimensional Gaussians of
Similarly to the first section we use the notation hri1,2 (t) ≡ d3 r ρ1,2 (r, t) r.
semiaxes a1 < b1 and a2 < b2 is another Gaussian whose semiaxes a < b
The average of (ρ1 ◦ (−1)) ⊗ ρ2 is the difference of the averages of ρ1 and
satisfy a1 + a2 < a < b < b1 + b2 . Therefore, given two error Gaussians
ρ2 :
Z
Z
of ellipsoids A1 < B1 < C1 and A2 < B2 < C2 , the projection of the error Gaussian of the relative position onto any plane is an ellipse of semiaxes
d2 ∆r ∆r d2 r1 ρ1 (r1 ) ρ2 (r1 + ∆r) =
a < b such that A1 + A2 < a < b < C1 + C2 .
Z
Z
(14)
We take the combined radius as the unit of length (this is, R1 +R2 = 1).
d2 ∆r ((∆r + r1 ) − r1 ) d2 r1 ρ1 (r1 ) ρ2 (r1 + ∆r) =
Based on the values reported in [16, 19, 27], we take the semiaxes of the error ellipsoids to range between 64 and 2048 along the velocity direction, and
hri2 − hri1 .
between 2 and 128 along the directions perpendicular to the velocity. Clearly
In the choice (ρ01 (r1 ), ρ02 (r2 )) = (δ(r1 + (hri2 − hri1 )), ((ρ1 ◦
the projections of the error ellipsoids of each object are ellipses whose semi(−1)) ⊗ ρ2 )(r2 + (hri2 − hri1 ))) one of the spheres is fixed at the location
axes range between 2 and 2048. Their convolution is an ellipse of semiaxes
hri1 − hri2 , while the other sphere is, on the average, at the origin but has
between 4 and 4096. We now answer the following question: Can each of
gathered all the position uncertainty of the two original spheres. This choice
the semiaxes of the ellipse take any value between 4 and 4096 independently
yields the following simplification:
of the value of the other semiaxis? It could be that there is no way to obtain,
say, an ellipse of semiaxes 4096 and 4096 by a projection and a convolution
i
∞
X i ∂ 2i ρ(hri1 − hri2 )
X R1 + R2 2i+2
4π
of the error ellipsoids. We shall see that this is actually the case and that only
,
p=
some combinations of the values of the semiaxes are possible.
2
(i + 1)(i!)2 j=0 j
∂x2(i−j) ∂y 2j
i=0
Let v1 and v2 be the expected velocities (in an inertial frame of origin
(15)
at the Earth’s center) of the objects which might collide. The long axes of
where ρ ≡ (ρ1 ◦ (−1)) ⊗ ρ2 .
the error ellipsoids are parallel to v1 and v2 . These velocities determine
the encounter plane, which is perpendicular to v1 − v2 . The intersection
4. THE GAUSSIAN CASE
between the plane of Fig. 1 and the encounter plane has been labeled lp. We
shall call sp the direction of the encounter plane perpendicular to lp. Then
When both ρ1 and ρ2 are Gaussians, (ρ1 ◦ (−1)) ⊗ ρ2 is a Gaussian G
the projections of the longer error ellipsoid semiaxes are always along lp, as
whose covariance matrix is the sum of the covariance matrices of ρ1 and
shown in the figure. Thus the projections along lp range from 2 to 2048, while
ρ2 and whose average is the difference of their averages. We choose the
the projections along sp range from 2 to 128. Therefore the error ellipse of
principal axes of G as axes of coordinates. In these axes let σx and σy be the
the relative positions has one semiaxis which ranges from 4 to 4096 (lp) and
standard deviations and let (x0 , y0 ) be the coordinates of r0 ≡ hri1 − hri2 .
another semiaxis which ranges from 4 to 256 (sp). In order to sample this
Then
region we go from 4 to 4096 and from 4 to 256 in powers of 4. This gives the
set 4, 16, 64, 256, 1024, 4096 for the first case and the set 4, 16, 64, 256 for
H2(i−j) (x0 /σx ) H2j (y0 /σy )
∂ 2i G(r0 )
G(r0 ),
(16)
=
the second. This gives 18 different error ellipses.
2(i−j) 2j
∂x2(i−j) ∂y 2j
σx
σy
where
G(r0 ) =
1
exp
2πσx σy
(
−
1
2
x20
y2
+ 02
2
σx
σy
!)
(17)
and Hn is the n-th probabilists’
polynomial,
defined by
n 2 o Hermite
n
o
dn
x2
Hn (x) = (−1)n exp x2 dx
[24, Chapter 8: Special
n exp − 2
functions] (not to be confused with the physicists’ Hermite polynomial [21,
Chapter 2: The classical orthogonal polynomials]; see also the Wikipedia
article “Hermite polynomials” 1 ).
Finally,
∞ ∞
X
X
R1 + R2 2i+2
4π
pi =
p=
2
(i
+
1)(i!)2
i=0
i=0
(18)
i X
i H2(i−j) (x0 /σx ) H2j (y0 /σy )
G(r
),
0
2(i−j) 2j
j
σx
σy
j=0
lp
v₂
v₁
where pi is the i-th term.
5. REGION OF INTEREST AND ITS SAMPLING
The more intuitive way to think is that pdf’s of the errors in the position
have to be convoluted and then projected on the encounter plane. But convolution and projections commute (see [25, section 9] or [26, section 10]),
and from a geometrical point of view it is easier to think of the projection
first and then consider the convolution of the 2-dimensional projections. To
each Gaussian there is an ellipsoid naturally associated to its covariance matrix. The axes of this ellipsoid are parallel to the principal directions of the
matrix and their lengths are proportional to the square roots of the eigenvalues of the covariance matrix. We are going to speak about this ellipsoid rather than about its covariance matrix. The projection of an ellipsoid
of semiaxes A < B < C is an ellipse whose semiaxes a < b satisfy
1 https://en.wikipedia.org/wiki/Hermite_
polynomials, accessed on September 10, 2015
Fig. 1. The encounter plane is perpendicular to v1 − v2
(dashed segment).
We have found the range of ellipses on the encounter plane to which the
Eq. (18) giving the probability of collision is applied. But there are other
parameters in the formula: the coordinates of the relative position. Eq. (18)
is not applied to any possible combination of parameters but only to those
for which an evasive maneuver is a non obvious possibility. For combinations of parameters for which a collision is very unlikely it is not necessary
to compute the probability, since a first screening can be done using cruder
methods [28, 29, 30]. Therefore computations are necessary only when the
probability of collision may surpass some predetermined value at which an
evasive maneuver is performed.
These threshold values are usually between 10−3 and 10−5 (see [5, 6,
7]). These values depend on issues such as economic considerations, international space legislation and insurance companies’ fees. However, the computation of the collision probability is also done when the decision to make
an evasive maneuver has been taken [6]. When this happens the maneuver is
required to yield a very small collision probability. A wider range of collision
probabilities has also been considered before [31]. For these reasons we will
study the integer powers of 10 in the range 10−1 to 10−7 [31].
For each of these values of constant p (10−1 , 10−2 , 10−3 ,
10−4 , 10−5 , 10−6 , 10−7 ) there is a curve of relative positions (x0 , y0 ) for
which the probability of collision takes these values. This curve is symmetric
with respect to the x0 and y0 axes. We take four points on this curve as
shown in Fig. 2: two along the axes of the ellipse (x0 , y0 ) = (x01 , 0)
and (x0 , y0 ) = (0, y04 ) and two others for which x02 = 2x01 /3 and
x03 = x01 /3. At first sight this would yield 18 × 7 × 4 = 504 sample
points. However, for some of the error ellipses and values of p, the curve
described above is the empty set: no relative position can yield that collision
probability. In the end there are only 244 cases left.
The interested reader is referred to the tables 1-5 of [1] for the numerical
values for these cases.
y
y₀₄
when the last term is smaller than the admissible absolute error then the error
is much smaller (between 27 and 56, 000 times smaller) than the last term
neglected. This is very strong evidence for the case that the last term can be
used as a bound for the absolute error in the region of interest.
In all the considered cases, the first two terms of the series (Eq .(18))
were necessary and sufficient to guarantee a 10% tolerance, that is
p ≈ p0 + p1 =
(
!) y02
R1 + R2 2
1 x20
2
exp −
+
σx σy
2 σx2
σy2
2

 2


2
y0
x0
2
−
1
−
1
1  σx
σy
 R1 + R2


+

1 + 
,
2
σx2
σy2
2
and the absolute error is bounded by
(
!) y02
R1 + R2 4
1 x20
1
exp −
+
|p1 | =
σx σy
2 σx2
σy2
2
2
2
x0
y0
σ
−1
− 1
σ
x
y
+
.
σx2
σy2
(19)
(20)
7. COMPARISON WITH OTHER AUTHORS
⅓x₀₁
⅔x₀₁
x₀₁
x
Fig. 2. Selected points in each curve of constant probability.
6. NUMBER OF TERMS NECESSARY FOR A GIVEN ACCURACY
Serra et alii [16] provide an algorithm which gives the number of terms necessary for a given accuracy in the computation of p. Other authors [14, 15]
give simple recipes to estimate the necessary number of terms based on the
error ellipse semiaxes, the relative position and the combined radius.
The expansion in Hermite polynomials converges everywhere, because,
as remarked in section 3, the Taylor expansion of the Gaussian converges uniformly everywhere. There are, however, regions in which the terms change
sign and become very large. This demands a very high precision and is a
numerical drawback. (In Serra et alii [16] an ingenious method, the preconditioning, is presented to circumvent this problem). This happens, for
example, when the ellipse of the Gaussian distribution is much smaller than
combined body. But these regions of the parameter space are not regions of
practical interest.
In the computation of the collision probability the uncertainty of the input is large. The pdf’s of the position are assumed to be Gaussians for general
reasons related to the central limit theorem and because of the analytic properties of the Gaussians, but it is not known to what extent they really are
Gaussians. The spheres are, as stated in the introduction, nothing but quite
conservative approximations to the actual shapes of the objects which may
collide. Therefore we use a tolerance of 10%, i. e., we allow a relative error
of 0.1 in p. For example if a probability of 0.0001 is obtained, we assure that
p ∈ [0.00009, 0.00011] for the given input. In other words, the absolute
error is smaller than 0.0001/10. Since the spherical approximation to the
shape of the objects is so conservative, we know that the collision probability
is actually . 0.0001.
For the threshold probabilities p used in our sampling we are demanding
absolute errors of at most p/10. In the 244 examples which we have used
to sample the region of interest, if we stop summing the series (Eq. (18))
In order to compute p and the number of terms needed for a given accuracy,
different authors developed analytical and numerical methods [9, 13, 14, 15,
16]. Of these methods, only the ones by Chan (for the isotropic cases only),
by Serra et alii and by ourselves are completely analytic 2 . Of the last two, our
method is simpler to derive and can be applied for non Gaussian pdf’s of the
position. Only Serra et alii present a completely analytic way to determine
the number of terms used to find p with a given accuracy.
All these algorithms can compute p with arbitrarily high accuracy. If
computing speed were not an issue, one would choose the simplest or most
pleasing method from a mathematical point of view, and some comments in
that regard are written in the preceding paragraph. But from a practical point
of view, speed is the first concern. Consequently, comparing the speed of the
different methods is an important criterion according to which we shall compare them. Other authors have compared different methods using different
criteria [16, 31].
We implemented all the algorithms (Foster, Patera, Alfano, Chan, Serra,
and the proposed method) in Fortran 95, since in order to compare speed,
it is convenient to use a low-level computing language like Fortran or C.
Then, we compared the performance of the methods on an Intel(R) Core(TM)
i5-2467M CPU at 1.60GHz machine running Ubuntu 14.04 and the GNU
Fortran 4.8.3-2 compiler.
Figure 3 shows for each method the time needed to compute each of
the 244 cases described above with a relative error smaller than 10%, as
advertised in the previous section. Eq. (19) is the fastest in each of the 244
cases. Similar but slightly longer times are obtained using Serra’s method
and Chan’s method, but the latter does not achieve the required accuracy
in some of the cases (number 175, 176, 179, 180, 183 and 194 of tables 4
and 9 of [1]). An erratic behaviour can be seen in the computation time of
Alfano’s Method in Fig 3, which is caused by the disparity in performance of
the Fortran intrinsic function erf (Error function) for different values of the
input.
8. CONCLUSIONS
A series to compute the collision probability of two spheres under the assumptions of short encounter has been derived. It is valid for both Gaussian
and non-Gaussian distributions of the position. In the particular case of a
2 Note that when applying the algorithm 1 of Serra et alii [16], one must
correct the numerators in the c2 and c3 coefficients: they should be 12 and
144 (instead of 6 and 24, respectively).
1
10
Foster
Patera
Alfano
Chan
Serra
New Method
[6] Righetti, P., Sancho, F., Lazaro, D. and Damiano, A., “Handling of
conjunction warnings in EUMETSAT flight dynamics,” Journal of
Aerospace Engineering, Sciences and Applications, vol. 3, no. 2, pp.
39, 2011.
0
10
execution time per case [ms]
no. 4, 2008.
doi: 10.2514/6.2008-3409.
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Fig. 3. Comparison of the execution time for the 244 different cases using different methods.
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Gaussian distribution the use of Hermite polynomials yields a simple form
for the series.
The region in which the parameters of the problem (probability density function of the relative position, relative position of the spheres, size of
spheres) define cases of practical interes t has been carefully defined. A sampling set of 244 cases in this region has been chosen. On this sampling set a
comparison between our series and previous algorithms has been performed
for the Gaussian case. Our series is faster than any other algorithm in every
case. Numerical evidence strongly suggests that if the series for the Gaussian
case is truncated when the last term is smaller than the computed probability
times a tolerance of 0.1, then the last term is an upper bound for the error.
The series presented here can yield arbitrary precision. But from a practical point of view, this article presents very strong evidence for the case that,
given the current lack of precision in the knowledge of the error of the position of the colliding objects, its first two terms alone are sufficient for the
computation of the probability of collision; and the absolute value of its second term is an upper bound for the error made when using it.
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